$\overline{AC} = 4$ $\overline{BC} = {?}$ $A$ $C$ $B$ $4$ $?$ $ \sin( \angle ABC ) = \frac{ \sqrt{5}}{5}, \cos( \angle ABC ) = \frac{2\sqrt{5} }{5}, \tan( \angle ABC ) = \dfrac{1}{2}$
Solution: $\overline{AC}$ is the opposite to $\angle ABC$ $\overline{BC}$ is adjacent to $\angle ABC$ SOH CAH TOA We know the opposite side and need to solve for the adjacent side so we can use the tan function (TOA) $ \tan( \angle ABC ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\overline{AC}}{\overline{BC}}= \frac{4}{\overline{BC}} $ $ \overline{BC}=\frac{4}{\tan( \angle ABC )} = \frac{4}{\dfrac{1}{2}} = 8$